I am trying to solve a problem from this website that asks the following:
Prove that, for a sequence of holomorphic functions on a compact set, convergence in $L^2$ implies uniform convergence.
I am struggling with how to relate the $L^2$ norm to the fact that the functions are holomorphic. There is also a related question, which asks if the space of square-integrable holomorphic functions on a bounded region $U\subseteq\mathbb C$ is a Banach space. This seems like it would follow from showing that a sequence of functions that is "$L^2$-Cauchy" is uniformly Cauchy, but I'm not sure.